From Tensor Network Quantum States to Tensorial Recurrent Neural Networks

TL;DR

This paper demonstrates that tensor network states, specifically matrix product states, can be exactly represented by recurrent neural networks with linear memory updates, enabling efficient wave function evaluation and sampling.

Contribution

It introduces a novel RNN architecture that generalizes tensor network states to 2D lattices, supporting perfect sampling and efficient wave function encoding.

Findings

RNNs can exactly represent MPS with linear memory updates.

The proposed model supports polynomial-time wave function evaluation.

It encodes wave functions with significantly lower bond dimension than traditional MPS.

Abstract

We show that any matrix product state (MPS) can be exactly represented by a recurrent neural network (RNN) with a linear memory update. We generalize this RNN architecture to 2D lattices using a multilinear memory update. It supports perfect sampling and wave function evaluation in polynomial time, and can represent an area law of entanglement entropy. Numerical evidence shows that it can encode the wave function using a bond dimension lower by orders of magnitude when compared to MPS, with an accuracy that can be systematically improved by increasing the bond dimension.